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(C) The radius $r$ of a charged particle moving in a uniform magnetic field $B$ is given by the formula $r = \frac{mv}{qB}$.Since the kinetic energy $

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$2: 1$

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(C) The radius $r$ of a charged particle moving in a uniform magnetic field $B$ is given by the formula $r = \frac{mv}{qB}$.Since the kinetic energy $K$ is related to momentum $p$ by $K = \frac{p^2}{2m}$,we have $p = \sqrt{2mK}$.Substituting $p = mv$ into the radius formula,we get $r = \frac{\sqrt{2mK}}{qB}$.This shows that the radius is directly proportional to the square root of the kinetic energy: $r \propto \sqrt{K}$.Given that the new kinetic energy $K_n = 4K_0$,where $K_0$ is the initial kinetic energy.The ratio of the new radius $r_n$ to the original radius $r_0$ is $\frac{r_n}{r_0} = \sqrt{\frac{K_n}{K_0}} = \sqrt{\frac{4K_0}{K_0}} = \sqrt{4} = 2$.Therefore,the ratio is $2: 1$.

$A$ charged particle moves along a circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charged particle increases to $4$ times its initial value. What will be the ratio of the new radius to the original radius of the circular path of the charged particle?

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